This is my pretty stupid theory:
When Shellder and Slowpoke attach together, the Slowpoke evolves into Slowbro and the Shellder evolves into something else (the spiral shape thing). My theory is a reaction inside causes Slowpoke to evolve when near Shellder, and the same reaction causes Shellder to evolve when near Slowpoke. In the G/S Pokedex, it says that Shellder can fall off of Slowbro's tail and then the Slowbro becomes a Slowpoke again. This also means that the Shellder returns to normal; that's why you don't ever see a spiral-shape Shellder seperate from a Slowpoke.

It's a reaction, like when you mix isopropyl alcohol with a lit match, or nitroglycerine with turpentine, or sodium bicarbonate with citric acid, or... Ah, forget it. :-)
Here's my theory:
There's some chemical in Slowbro's tail... Not unlike citric acid... Which reacts with the inside of Shellder's shell, which is like sodium bicarbonate. When the two combine, it creates a new chemical, like the result of Sodium Bicarbonate and Citric Acid, which affects the two Pokémon. This chemical somehow sends signals to their brain, like a hormone, to make them evolve. Wow, I just freaked myself out...

Even if Shellder was capable of using transform, it would turn into a Slowpoke, because it cannot transform into Pokemon/objects on its own; it's gotta mirror something. In the episode where Ditto was first introduced, Ditto was able to turn into a cannon because Team Rocket was aiming a cannon at it. That's proof of how transform works.
Also, to again answer the slipup, I think the professor gave sufficient reason why Shellder turns spiral shelled, so I don't see credibility in this slip-up.

I think that the Shelder changes shape because the spiky shape is already underneath the smooth shell, you never saw the in side of a shelder did you? The old shell simply slides off when the Shelder is clamping up on Slowpoke's tail

Can a normal shelled Shellder clamp onto Slowpoke's tail? Of course not; it's too flat. The only way a Shellder can "fit" on Slowpoke's tail is if it becomes spiral shape. They never said this directly, but I think that a Shellder's "will" to go on Slowpoke's tail causes it to turn spiral. Shellder wants to get out of the water onto the land, Slowpoke can use moves that require it to go on its two front feet. Shellder's will must be strong. :)

That saying goes:
"I'm sorry Westwood the 1st, 2nd, 3rd, 4th, 5th... ohh I forgot thats me!"
He's only the 5th, not the 6th. But anyway heres anoter correction, Slowbro does NOT evolve into Slowking.
There are 2 ways I can answer how this happens (Thus proving I am smarter than prof. Westwood)
1, the gomeboy answer: Slowpoke evolves into Slowking, I know this for a fact because I just recently traded my Slowpoke holding a metal coat to myself on another game pak and "What? Slowpoke is evolving!" then... *TAH DAH* theres a Slowking.
2, the anime show answer: Slowpoke fishes for a shelldar and catches one, then he places it on his head before they can evolve together, thus making a Slowking.
As or this whole "How did shelldar fit?" and stuff like that, one word will solve this problem, EVOLUTION. Holy cow people, tomorrow you will come in here asking "How in the Poké world did Pikachu suddenly turn orange, grow a giant tail, and suddenly start saying Raichu?"

The episode was called "The Evolution Solution". And I think that Shellder becomes a spiral shape to make itself more streamlined when Slowbro is swimming. But that has nothing to do with this slip-up. The point is, the Professor doesn't know, even if we have our own possible theories.
Professor Westwood, even though he's a Pokémon scientist, is pretty stupid. "Aargh! I have angered Professor Westwood I, II, III, IV, V, VI! Oh no, wait a minute, that's me."

Hmm, I guess I read through these comments too fast, no one mentioned Slowking, but I got confused when you were talking about how Shelldar can fall off and stuff like that. In the show he might be able to do that but in the gameboy games that is impossible. This relates to my other comment because Slowbro can't pull that pesky Shelldar off his tail after it has evolved and place it on his head, he has to put the Shalldar on his noggin before they can evolve.